Derivatives 2022

Dr. Fabian Woebbeking | fabian.woebbeking@dozent.frankfurt-school.de

Content

Material

Reading

Introduction / a primer on arbitrage

Put-Call-Parity

Put-Call-Parity is the arbitrage relationship between (European) put, call, and forward.

$$PV(Call) - PV(Put) = S_0 - PV(K)$$

With cash flows at maturity $T$:

$CF(T, Call) = max(S_T - K, 0) = (S_T - K)^+$

$CF(T, Put) = max(K - S_T, 0) = (K - S_T)^+$

$CF(T, Fwd) = S_T - K$

For example, if a call option is overpriced w.r.t. its replicating portfolio, i.e. Fwd + Put, sell the call and buy the replication. The resulting shift in supply and demand must adjust prices.

Tulip Mania in Holland (17th century)

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Weapons of mass destruction?

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Heidorn, T.; Mokinski, F; Rühl, C.; Schmaltz, C.(2015): The impact of fundamental and financial traders on the term structure of oil Energy Economics 46 (2015) pp. 276 – 287

Symmetric derivatives

Interest rate languages

Interest rates are quoted per annum (p.a.) but speak different “languages”:

Money Market (MM):

$C_0(1+r_{mm}T_{mm})=C_T$

ISMA (International Securities Market Association - Europe)

$C_0(1+r_{isma})^T=C_T$

SIA (Securities Industry Association - US)

$C_0(1+r_{sia}/2)^{2T}=C_T$

Continous compounding (academia, financial engineering)

$C_0 e^{rT} = C_T$

T is calculated by dividing the number of days in perid T by a year basis, e.g.

$T = act/360$ (Money Markets - DE/US/...)

$T = act/act$ (Bond - ISMA)

$T = 30/360$ (Swap)

These are some examples for day count conventions, which might differ from country to country. Observe that for $act/act$ and $30/360$, a full year must always yield $T=1$.

Semi-annual compounding has value only for didactical puposes, this is, when increasing the number of compounding periods $m$

$$C_0 \left( 1+\frac{r}{m} \right)^{mT}=C_T$$

Forward valuation (cost of carry)

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To replicate a forward consider a portfolio that consists of underlying $S$ and cash $K$, which today is worth

$$f_0 = S_0 - K e^{-rT}$$

and at maturity $T$

$$f_T = S_T - K$$

The forward rate $F$ at initiation is set such that $f_0 = 0$, hence,

$$S_0 - F e^{-rt} = 0 \Leftrightarrow F = S e^{rT}$$

For many assets, additional factors affect the forward price, such as:

Considering all factors, we have

$$f_0 = S_0 e^{(-q-y+u)T} - K e^{-rT}$$

and

$$F = S e^{(r-q+u-y)T}$$

Basis risk

The risk of imperfect hedging.

Consider a long position in a risky asset $S$ to be hedged by selling (short) future $F$. We will hedge the asset in $t=1$, close the hedge in $t=2$ and analyze the basis $b$:

$b_1 := S_1 - F_1$

$b_2 := S_2 - F_2$

Value of the portfolio in $t=2$

$S_2 + F_1 - F_2 = \underbrace{F_1 + b_2}_{b_2 = S_2 - F_2}$

$b_2$ is unknown in $t=1$, hence, risky. Now also consider the underlying of the future $S^*$

$S_2 + F_1 - F_2 = F_1 + \underbrace{(S^*_2 - F_2)}_\text{time basis} + \underbrace{(S_2 - S^*_2)}_\text{underlying basis} \hspace{1cm} | S^*_2 - S^*_2 $

Reasons for basis risk:

Optimal hedge ratio

Hedge ratio for hedging asset S with future F

$$Var = \sigma^2_S + h^2\sigma^2_F-2 h \rho \sigma_S \sigma_F$$$$\frac{\delta Var}{\delta h} = 2 h \sigma^2_F - 2 \rho \sigma_S \sigma_F = 0$$$$...$$$$h = \rho \frac{\sigma_S}{\sigma_F}$$

Bonds

A discount factor is today's price for 1 monetary unit in t, hence, a zero bond with face value 1 and maturity t.

$$C_0 = C_t DF_t \implies DF_t = \frac{C_0}{C_t}$$

The discount factor does not speak a language! However, it can be translated into any interest rate language, e.g.

$$C_0 = C_t \frac{1}{(1 + r)^t} \implies DF_t = \frac{1}{(1+r)^t} \implies r = \left(\frac{1}{DF_t}\right)^{1/t} - 1$$

And as usual

$$PV = \sum^{T}_{t=1}\frac{C_t}{(1+r_t)^t} = \sum^{T}_{t=1}C_t DF_t$$

Reference rates (IBOR reform)

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Interest rate swaps

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Liability swap

Liability Swap: Receive(Swap) - Pay(effective) = Financing Benefit

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Asset swap

Asset Swap: Receive(Effective) – Pay(Swap) = Investment Benefit

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Discount factors and forward rates

Forward rate fixed in $s$ payed in $t$ has arbitrage condition

$$(1+r_s)^s(1+r_{st})^{t-s} = (1+r_t)^t$$$$r_{st} = \left(\frac{(1+r_t)^t}{(1+r_s)^s}\right)^{1/(t-s)}-1$$

Again, a discount factor $DF_t$ is today's value of 1 that is paid in $t$. Conversely, $1 / DF_t$ is a future value. This allows us to calculate "forward discount factors" by discounting $DF_t$ and compounding a shorter period $s$ with $1/DF_s$. Discounting with a forward discount factor or a forward rate must yield the same, hence,

$$\begin{aligned} \frac{DF_t}{DF_s} &= \frac{1}{(1 + r_{st})^{t-s}}\\ &... \\ r_{st} &= \left(\frac{DF_s}{DF_t}\right)^{1/(t-s)}-1\\ &= \left(\frac{(1+r_t)^t}{(1+r_s)^s}\right)^{1/(t-s)}-1 \end{aligned}$$
Plain vanilla swap pricing

Agreement to exchange a fixed ($C$) against a variable ($L_t$) payment, assuming annual payments

$$\begin{aligned} PV(Fix) &= C \sum_{t=1}^T DF_t\\ PV(Var) &= \sum_{t=1}^T L_t DF_t\\ &= \sum_{t=1}^T \left(\frac{DF_{t-1}}{DF_t} -1 \right) DF_t\\ &= 1 - DF_T \end{aligned}$$

Price the fair swap rate $C$ such that,

$$\begin{aligned} PV(Fix) &= PV(Var)\\ C \sum_{t=1}^T DF_t &= 1 - DF_T\\ C &= \frac{1 - DF_T}{\sum_{t=1}^T DF_t} \end{aligned}$$

Mark to market an existing swap with $C^*$ (the old swap rate) by comparing $PV(Fix)$ and $PV(Var)$

$$\begin{aligned} PV(Receiver) &= PV(Fix) - PV(Var)\\ &= C^* \sum_{t=1}^T DF_t - (1 - DF_T) \\ &= (C^* - C) \sum_{t=1}^T DF_t \end{aligned}$$

Swap rate $C$ is also the par coupon bond rate that fulfills

$$\begin{aligned} 1 &= \sum^{T}_{t=1}C_t DF_t\\ &= C \sum^{T}_{t=1}DF_t + DF_T \implies C = \frac{1 - DF_T}{\sum^{T}_{t=1}DF_t} \end{aligned}$$

Single curve approach

Use one curve to

  1. calculate (bootstrap) spot rates (discount factors)
  2. calculate forward rates (substitute $L_t$ as cash flow)
  3. solve for $C$ such that $PV(Fix)$ and $PV(Var)$

Bootstrapping

Bootstrapping, i.e. creating zero bonds from a market of coupon bearing instruments

$$DF_t = \frac{1 - C_t \sum_{i=1}^{t-1} DF_i}{1 + C_t}$$

Multi curve approach

Swap market after the financial crisis

Since the financial crisis exists a considerable spread between 6M Euribor and 6M Eonia Swap

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Since there exists a basis, it can be traded (basis swap).

Single curve

Multi curve (implied forwards)

Implied forward rate $$\tilde{L_T} = \frac{C_T \sum_{t=1}^T DF_t - \sum_{t=1}^{T-1} L_t DF_t}{DF_T} \equiv \frac{CD - LD}{DF_T}$$

Note that

$$\begin{aligned} PV(Fix) &=PV(Var)\\ C_T \sum_{t=1}^T DF_t &= \sum_{t=1}^T \tilde{L_t} DF_t = \sum_{t=1}^{T-1} \tilde{L_t} DF_t + \tilde{L_T} DF_T\\ \end{aligned}$$

Cross currency swap (CCY)

CCY basis swap:

  1. Exchange of interest rates in different currencies
  2. Nominal exchange with todays FS spot rate
  3. Agreement on FX forward for the nominal at the original spot FX rate

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In theory the spread of the product should be almost zero. However due to different liquidity levels and different credit levels in the currencies in reality we find substantial spreads.

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FX Swap

Let us start with two definitions:

$$\text{EUR|USD} = 1.3591$$$$\text{USD|JPY} = 82.407$$

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$$\text{EUR|JPY} = \text{EUR|USD} \times \text{USD|JPY} = 1.3591 \times 82.407 = 111.999$$$$\text{EUR|GBP} = \text{EUR|USD} \times \text{GBP|USD}^{-1} = 1.3591 \times \frac{1}{1.6136} = 0.84228$$

The FX swap is an agreement to exchange spot at $S$ as well as in the future at forward rate $F_T$. According to Covered Interest Rate Parity, this has arbitrage relationship:

$$ \begin{align*} (1 + r_B \times T_B) F_T &= S (1 + r_Q \times T_Q) \\ &...\\ F_T &= S \frac{(1 + r_Q \times T_Q)}{(1 + r_B \times T_B)} \end{align*} $$

By convention, FX forwards (aka FX ourights) are often quotet in FX swap points, this is,

$$\text{FX swap points} = F_T - S$$

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The FX swap market includes besides the interest rate differential also liquidity and credit advantages of the two currencies. Since the financial crisis the interest differential (interest rate parity) is not a good indicator for the FX swap anymore.

Forward rate agreement (FRA)

A Forward Rate Agreement (or FRA) is an agreement between two parties to exchange a short term underlying interest rate in the future.

$$\text{Settlement amount} = \frac{(L(d_2)-r(d_3,d_4)) \frac{act}{360}}{\left(1+L(d_2) \frac{act}{360}\right)} \times \pm \text{Nominal}$$

A buyer enters into the contract to protect himself from increasing interest rates, hence, $+ \rightarrow$ buyer, $- \rightarrow$ seller.

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Future clearing, settlement and hedging

We will use a Bund future to highlight the peculiarities of future clearing, settlement and hedging. The Bund future is a future traded at EUREX on a hypothetical long term German government bond.

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Clearing

A clearing mechanism is a margining mechanism the involves one or more of:

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Settlement

Bund future specifications:

The ideal bond is not available, thus, we need a conversion factor $\text{CF}_{\text{Bond}}$

$$ \begin{align*} \text{CF}_{\text{Bond}} &= \frac{\text{PV}_{\text{Bond}}(@6\%)}{\text{PV}_{\text{Underlying}}(@6\%)}\\ &= \frac{\text{PV}_{\text{Bond}}(@6\%)}{100}. \end{align*} $$

The conversion factor allows us to calculate the payment at deilvery

$$\text{Delivery payment} = \text{EDSP} \times \text{CF}_{\text{Bond}} \times 100,000 + \text{Accrued} $$

“The final settlement price [EDSP] is established by Eurex on the final settlement day at 12:30 CET based on the volume-weighted average price of all trades during the final minute of trading provided that more than ten trades occurred during this minute; otherwise the volume-weighted average price of the last ten trades of the day, provided that these are not older than 30 minutes. If such a price cannot be determined, or does not reasonably reflect the prevailing market conditions, Eurex will establish the final settlement price.” (Eurex Exchange 2019)

Cheapest-to-deliver

Now that we have the conversion factor ($\text{CF}_{\text{Bond}}$), we have to choose which bond to deliver at maturity. This is the cheapest-to-deliver, which can be identified by its implied repo rate

$$ \text{Implied repo} = \frac{\text{Delivery payment} - \text{Dirty price}}{\text{Dirty price} \times \frac{\text{Days to settlement}}{360}} $$

A Repo trade is the agreement to sell (Spot) and buy back ($t+n$) an asset. Spot short and at the same time future long can be seen as a repo trade. Another interpretation is collateralized lending. The bond with the highest Implied Repo rate is the cheapest to deliver (CTD).

Theoretical future price

The forward price for the bond cheapest to deliver (CTD) is determined by the costs of acquiring the bond now and holding it until it shall be delivered to fulfil the obligation from the future. The cost of carry model applies, hence,

$$\text{Forward price} = \text{Spot} + \text{Financing} - \text{Accrued}$$

Considering the underlying of the future,

$$\text{Theoretical future} = \frac{\text{Forward price}}{\text{CF}_{\text{Bond}}}$$

Hedging with bund futures

$$\text{Simple hedge ratio} = \frac{\text{Nominal spot position}}{\text{Nominal future}}$$

Basis Point Value (BPV = PVO1) Hedge is a Nominal hedge, adjusted for underlying Basis Risk.

$$\text{Basis point hedge ratio} = \text{Simple hedge ratio} \times \text{CF}_{\text{Bond}} \times \frac{\text{BPV}_\text{Spot}}{\text{BPV}_\text{CTD}}$$

PVO1 or BPV is the MTM change caused by a $0.0001$ = 1bp shift in yield.

Asymmetric derivatives

Assumptions

Option price boundaries

Notation:

Lower boundaries       Upper boundaries
European Call $c \ge S-K e^{-rT}$ $c \le S$
American Call $C \ge S-Ke^{-rT}$ $C \le S$
European Put $p \ge K e^{-rT} - S$ $p \le K e^{-rT}$
American Put $P \ge K - S$ $P \le K$

Why it makes no sense to execute an american call option prematurely:

Probability distributions

Consider random variable $X \sim \mathcal{N}(\mu,\sigma^2)$ as an example for an absolutely continuous univariate distribution.

Probability density function (PDF): $$\phi(x) = \frac{d\Phi(x)}{dx} = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 }$$

Cumulative distribution function (CDF): $$\Phi(x) = P(X \le x) = \int_{-\infty}^x \phi(u)\; du$$

Quantile function, i.e. inverse of the CDF: $$\Phi^{-1}(p) = \inf \{x \in \mathbb{R}: p \le \Phi(x)\},\;\; p \in (0,1)$$

Stochastic processes

Markov property

In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process, i.e. only the present price is relevant for predicting the future. Markov property is consistent with the weak form of market efficiency!

Generalized Wiener process

A stochastic process $X$, with drift $\mu$ and dispersion $\sigma$, follows a generalized Wiener process if it satisfies the following stochastic differential equation (SDE)

$$dX_t = \mu \times dt + \sigma \times dW_t$$

Note that by definition, the increments of a wiener process $W_t = W_t - W_0 \sim \mathcal{N}(0,t)$, i.e. are normally distributed, centered at zero.

For an arbitrary initial value $X_0$

$$X_t = X_0 + \mu \times t + \sigma \times z \times \sqrt{t}$$

, where $z \sim \mathcal{N}(0,1)$.

Geometric Brownian motion

A stochastic process $S$ follows a Geometric Brownian Motion (GBM) if it satisfies the following stochastic differential equation (SDE)

$$dS_t = \mu S_t dt + \sigma S_t dW_t$$

where $W_t = W_t - W_0 \sim \mathcal{N}(0,t)$, i.e. a Wiener process, hence, normally distributed increments.

For an arbitrary initial value $S_0$, the above SDE has the following analytical solution (under Ito's interpretation)

$$S_t = S_0 \exp\left( \left(\mu - \frac{\sigma^2}{2}\right) t + \sigma W_t \right)$$

, hence,

$$\ln \frac{S_t}{S_0} = \left(\mu - \frac{\sigma^2}{2}\right) t + \sigma W_t$$

Note that $W_t = z \sqrt{t}$, where $z \sim \mathcal{N}(0,1)$. Therefore, $\ln \frac{S_t}{S_0}$ (log-return) is normally and $S_t$ is log-normally distributed.

Option pricing via Monte Carlo simulation

Binomial option pricing models

Consider a replicating portfolio with bond (cash) $B$ and $\Delta$ share in the underlying asset $S$

$$ \begin{align} V_0 &= B + \Delta S_0\\ V_u &= B e^{rt} + \Delta S_u = (S_u - K)^+\\ V_d &= B e^{rt} + \Delta S_d = (S_d - K)^+ \end{align} $$

Note that

$$\Delta = \frac{(S_u - K)^+ - (S_d - K)^+}{S_u - S_d}$$

Example:

$$ \begin{align} V_u &= -28.54 e^{0.5} + 0.75 \times 60 = 15\\ V_d &= -28.54 e^{0.5} + 0.75 \times 40 = 0\\ &...\\ V_0 &= -28.54 + 0.75 \times 50 = 8.96\\ \end{align} $$

Coxx-Ross-Rubinstein (CRR)

The underlying of an option with maturity $T$ is modelled by an $n$ step binomial tree, with time between two steps of the tree $\Delta t = T / n$. With up move $u$ and down move $d$

$$u = \exp\left(\sigma \sqrt{\Delta t}\right)$$$$d = u^{-1}$$

The following must hold for the up move probability $p$, in order to ensure martingale property,

$$S_0 = \left( S_0 \times u \times p + S_0 \times d \times (1-p) \right) e^{-r\Delta t} \implies p = \frac{e^{r\Delta t} - d}{u - d}$$

Remember the binomial probability mass function (PMF)

$$f(k,n,p) = {n \choose k} p^k (1-p)^{n-k}$$

here, $n$ is the number of steps in the tree and $k$ the number of up moves to reach a state $S_T$.

European vs American options

The CRR model is solved recuresively by discounting expected values. For American options, the buyer has the right to exercise at any point in time. Due to its discrete modelling of time, the CRR model allows execution at any node in the lattice. However, the buyer would only execute if the value that can be realized through execution $IV_t$ (intrinsic value) is above the discounted expected value of the next node. Therefore, the value that is considered for each node is

$$V_t = \max\left(E[V_{t+\Delta t}]e^{-r\Delta t}, IV_t \right).$$

CRR solution

Solve recursively by stepwise discounting $V_t$ backwards in the tree. In this example we consider a European call with $S_0 = 220$, $K=165$, $\sigma = 98\%$, $T=1$, $r = 21\%$ ISDA.

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Bianary options

The Black-Scholes-Merton (BSM) model class

Black Scholes (no dividend)

$$PV(Call) = S_0 N(d_+)− K e^{-rT} N(d_-)$$$$PV(Put) = K e^{-rT} N(-d_-) − S_0 N(-d_+)$$

with

$$d_\pm = \frac{\ln\left(\frac{S_0}{K}\right) + rT \pm \sigma^2 \frac{T}{2}}{\sigma \sqrt{T}}$$

Merton (incl. dividend)

$$PV(Call) = S_0 e^{-qT} N(d_+)− K e^{-rT} N(d_-)$$$$PV(Put) = K e^{-rT} N(-d_-) − S_0 e^{-qT} N(-d_+)$$

with

$$d_\pm = \frac{\ln\left(\frac{S_0}{K}\right) + (r-q)T \pm \sigma^2 \frac{T}{2}}{\sigma \sqrt{T}}$$

Black76 (forward)

$$PV(Call) = \left[F_T N(d_+)− K N(d_-)\right] e^{-rT}$$$$PV(Put) = \left[K N(-d_-) − F_T N(-d_+) \right] e^{-rT}$$

with

$$d_\pm = \frac{\ln\left(\frac{F_T}{K}\right) \pm \sigma^2 \frac{T}{2}}{\sigma \sqrt{T}}$$

Option pricing via straightforward integration (BSM)

European call option

The European call price is simply the discounted expected value of its payout $E^\mathbb{Q}[(S_T - K)^+] e^{-rT}$, with

$$\begin{align} E^\mathbb{Q}[(S_T - K)^+] &= \int_K^\infty (S_T - K) f(S_T) dS_T\\ &=\int_K^\infty S_T f(S_T) dS_T - K \int_K^\infty f(S_T) dS_T \end{align}$$

The terminal stock price at time $T$, under the risk-neutral measure, follows a log-normal distribution with $\mu = \ln S_0 + (r_f-\sigma^2 / 2)T$, variance $s^2 = \sigma^2 T$ and PDF

$$f(x) = \frac{1}{x s \sqrt{2 \pi}} \exp\left( -\frac{(\ln x - \mu)^2}{2 s^2} \right) $$

Binary options

Asset-or-nothing option

Consider a contract where you receive the underlying $S_T$ conditional on $S_T > K$, note that this is different from a call option where the cash flow is $S_T - K$ under the condition that $S_T > K$. The price of this (expensive) option is $E^\mathbb{Q}[S_T 1_{\{S_T > K\}}] e^{-rT}$, where

$$E^\mathbb{Q}[S_T 1_{\{S_T > K\}}] = \int_K^\infty S_T f(S_T) dS_T$$

Cash-or-nothing option

Now consider an option that pays some amount $K$ (cash) conditional on $S_T > K$. The price of this (again expensive) option is $K \times P^\mathbb{Q}[S_T > K] e^{-rT}$, where

$$P^\mathbb{Q}[S_T > K] = \int_K^\infty f(S_T) dS_T$$

Therefore, a European call option is a portfolio out of a long asset-or-noting option and a short cash-or-nothing option.

Greeks

Greeks are partial derivatives that describe the sensitivity of the price of an option to a change in the respective underlying parameters.

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Greeks are not constant, see e.g. delta (red) and gamma (blue)

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Implied volatility

All parameters that determine the option price in a BSM model are observable, except $\sigma$. However, option prices are not calculated, they are the result of supply and demand on a liquid market. Thus, given an option price, we can find the $\sigma$ that ceteris paribus equates the BSM price with the market price.

Consider a matrix of option prices for different strikes and maturities. According to the theory, the resulting volatility surface should be flat, however, we usually observe sth like

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Reasons for a non-flat volatility surface

Credit risk and credit derivatives

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Credit default swaps (CDS)

A CDS contract works like an insurence that is linked to a credit event. This is, the protection buyer pays a premium $S$ (CDS spread) to the protection seller until the contract matures in $T$. In case of a default during the term of the contract, i.e. default in $\tau < T$, the protection buyer receives a settlement payment from the seller. This settlement payment is often, but not necessarily, designed to cover the loss given the default of a credit risky position, i.e. $(1 - \mathrm{Recovery})$.

Credit events (ISDA credit derivative definition 5/99 6/03 4/09 2/14):

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Replication

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The payout can be replicarted with one credit risky and another risk free floater, which both trade at par. Shorting the risky floater yields 100% nominal, which can be invested into the risk free floater. The latter pays per definition the risk free rate $r_f$, wheres payments into the risky floater include an additional (credit) spread $r_f + S_{FRN = Par}$. In case of a credit event, the risk free floater is sold at 100%, whereas the risky floater is bought back at recovery. As this replicates the payout of the CDS $(1 - \mathrm{Recovery})$, the replicating credit spread must be equal to the CDS spread $S$. However, this is only an approximation, as it ignores that recovery is only paid on the nominal amount, transaction costs, and liquidity differences (availability) of the floaters.

$$S_{FRN = Par} \approx S$$

If a floater is not trading at par, the replication does not work. However, the PV difference must be due to the discounted difference in spreads. Thus, given the spread $S_{FRN \neq Par}$ and price $P_{FRN\neq Par}$ of a non par floater, we can solve for the spread of the corresponding par floater $S_{FRN = Par}$

$$ \begin{align} P_{FRN\neq Par} - 100\% &= PV(S_{FRN\neq Par} - S_{FRN = Par})\\ &= (S_{FRN\neq Par} - S_{FRN = Par}) \sum_{t=1}^T DF_t\\ &...\\ S_{FRN = Par} &= S_{FRN\neq Par} - \frac{P_{FRN \neq Par} - 100\%}{\sum_{t=1}^T DF_t} \end{align} $$

In reality, costs arise from shorting the FRN, i.e. the bond has to be borrowd for a fee $S_{Repo}$. Thus

$$S_{FRN = Par} + S_{Repo} = S$$

No floater available? No problem, use an asset swap to create a synthetic floater, which at par is worth

$$ \begin{align} 100\% &= P_\mathrm{Bond} - PV(C_\mathrm{Bond}) + PV(\mathrm{Libor} + S)\\\\ &= P_\mathrm{Bond} - C_\mathrm{Bond} \sum_{t=1}^T DF_t + 1 - DF_T + S \sum_{t=1}^T DF_t\\\\ S &= \frac{DF_T + C_\mathrm{Bond} \sum_{t=1}^T DF_t - P_\mathrm{Bond}}{\sum_{t=1}^T DF_t} \end{align} $$

If the bond is trading at par, $S$ is simply

$$S = C_\mathrm{Bond} - \frac{1-DF_T}{\sum_{t=1}^T DF_t}\\\\$$

The asset swap spread is only an approximation for the CDS spread, as the CDS cash flow is not perfectly replicated (close out value of the swap) and there might be liquidity issues.

Valuation with hazard rates

As always, the price of a financial product is the sum of its discounted (expected) cash flows. Accounting for credit risk simply means accounting for both, (risk free) interest rate and the probability of default.

Let $h_t$, the hazard rate (default intensity), be the probability to default in an infinitesimally short time interval $dt$.

$$h_t = \ln\left(\frac{1-p_{t-1}}{1-p_t} \right) \implies e^{h_t} = \frac{1-p_{t-1}}{1-p_t} \Longleftrightarrow p_t = 1 - \frac{1- p_{t-1}}{e^{h_t}}$$

Then $a_t(h_t)$ is the present value of a paymment of $1$ at time $t$ if $t < \tau$, hence, no default. Think of this as a discount factor ($DF_t$) that also accounts for the likelyhood of the asset being alive.

$$a_t(h_t) = e^{-(h_t+r_t)t}$$

On the other hand, $b_t(h_t)$ is the present value of a payment of 1 at time $t$ if $t-1 < \tau < t$, hence, default between $t-1$ and $t$.

$$b_t(h_t) = e^{-r_t t} \left( e^{-(h_t)t_{-1}} - e^{-(h_t)t} \right)$$

Summing up the risky discount factors $a_t(h_t)$ and $b_t(h_t)$

$$ \begin{align} A_T &= \sum_{t=1}^T a_t\\ B_T &= \sum_{t=1}^T b_t \end{align} $$

Allows us to price a CDS with spread $S$ as

$$ \begin{align} 0 &= PV(1-\mathrm{Recovery}) - PV(S)\\ &= B_T(1-\mathrm{Recovery}) - A_T S\\\\ S &= \frac{B_T(1-\mathrm{Recovery})}{A_T} \end{align} $$

Hint: compare this to the single curve swap pricing above.

In practise, in preparation for central counterparty clearing, spread payments into a CDS are fixed at, i.e. $S^* \in \{0.002,0.01,0.05,0.1\}$, see ISDA agreements. Also the recovery rate is fixed, e.g. to 40% for investment grade. Therefore, contracts are unfair when entered and require an upfont payment

$$\mathrm{Upfront} = B_T(1-\mathrm{Recovery}) - A_T S^*$$

Hazard rate calibration

Recall the CDS replication via floating rate notes, solve for $h$ in $A_T$ and $B_T$, using that $S_{FRN = Par} \approx S$

$$ \begin{align} P_{FRN} - 100\% &= (S_{FRN} - S) A_T\\ &= A_T S_{FRN} - B_T(1-\mathrm{Recovery})\\ \end{align} $$

Note that

$$\frac{S}{1-\mathrm{Recovery}} = \frac{B_T}{A_T} \approx h$$

This approximation is often called "credit triangle".

Thank you for your attention

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